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Recamán's sequence

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In mathematics and computer science, Recamán's sequence[1][2] is a well known sequence defined by a recurrence relation. Because its elements are related to the previous elements in a straightforward way, they are often defined using recursion.

A drawing of the first 75 terms of Recamán's sequence, according with the method of visualization shown in the Numberphile video The Slightly Spooky Recamán Sequence[3]

It takes its name after its inventor Bernardo Recamán Santos [es], a Colombian mathematician.

Definition

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Recamán's sequence is defined as:

The first terms of the sequence are:

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, 42, 63, 41, 18, 42, 17, 43, 16, 44, 15, 45, 14, 46, 79, 113, 78, 114, 77, 39, 78, 38, 79, 37, 80, 36, 81, 35, 82, 34, 83, 33, 84, 32, 85, 31, 86, 30, 87, 29, 88, 28, 89, 27, 90, 26, 91, 157, 224, 156, 225, 155, ...

On-line encyclopedia of integer sequences (OEIS)

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Recamán's sequence was named after its inventor, Colombian mathematician Bernardo Recamán Santos, by Neil Sloane, creator of the On-Line Encyclopedia of Integer Sequences (OEIS). The OEIS entry for this sequence is A005132.

Visual representation

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A plot for the first 100 terms of the Recamán's sequence.[4]

The most-common visualization of the Recamán's sequence is simply plotting its values, such as the figure at right.

On January 14, 2018, the Numberphile YouTube channel published a video titled The Slightly Spooky Recamán Sequence,[3] showing a visualization using alternating semi-circles, as it is shown in the figure at top of this page.

Sound representation

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Values of the sequence can be associated with musical notes, in such that case the running of the sequence can be associated with an execution of a musical tune.[5]

Properties

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The sequence satisfies:[1]

This is not a permutation of the integers: the first repeated term is .[6] Another one is .

Conjecture

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Neil Sloane has conjectured that every number eventually appears,[7][8][9] but it has not been proved. Even though 10230 terms have been calculated (in 2018), the number 852,655 has not appeared on the list.[1]

Uses

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Besides its mathematical and aesthetic properties, Recamán's sequence can be used to secure 2D images by steganography.[10]

Alternate sequence

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The sequence is the most-known sequence invented by Recamán. There is another sequence, less known, defined as:

This OEIS entry is A008336.

References

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  1. ^ a b c "A005132 - Oeis".
  2. ^ "Recamán's Sequence".
  3. ^ a b The Slightly Spooky Recamán Sequence, Numberphile video.
  4. ^ R.Ugalde, Laurence. "Recamán sequence in Fōrmulæ programming language". Fōrmulæ. Retrieved July 26, 2021.
  5. ^ "The On-Line Encyclopedia of Integer Sequences® (OEIS®)".
  6. ^ Math less traveled
  7. ^ "A057167 - Oeis".
  8. ^ "A064227 - Oeis".
  9. ^ "A064228 - Oeis".
  10. ^ S. Farrag and W. Alexan, "Secure 2D Image Steganography Using Recamán's Sequence," 2019 International Conference on Advanced Communication Technologies and Networking (CommNet), Rabat, Morocco, 2019, pp. 1-6. doi: 10.1109/COMMNET.2019.8742368
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